Beyond Nature

India:
When we hear this name, the first thing that strikes our mind is population.........
then if people are asked what is india famous for.....?
these will be the answers....
Poverty,Illiteracy,Cricket,Taj Mahal, movies , tourism etc.........
but atleast one will be there who tells MATHEMATICS,SCIENCE,ASTROLOGY,CULTURE,CIVILIZATION....
he is the true indian..(he had recognized the greatness of our country)...

Do you know that ........?
earth's diameter was first calculated by an INDIAN
pie value was evaluated by an INDIAN
exact value of square root of 2 was given by an INDIAN
geocentric theory was proposed by INDIANS
Trigonometry was developed by INDIANS
0 was given by INDIANS
and many more......
But people say INDIANS CONTRIBUTION TO MATHS IS ZERO (in both ways)...

show this to them and say
INDIANS have really gone BEYOND NATURE
lets now know about ARYABATTA (born genius) and his works...



Biography

Though Aryabhata's year of birth is clearly mentioned in Aryabhatiya, exact location of his place of birth remains a matter of contention amongst the scholars. Some scholars argue that Aryabhata was born in Kusumapura, while others argue that Aryabhata was from Kerala.[1]Some believe he was born in the region lying between Narmada and Godavari, which was known as Ashmaka and they identify Ashmaka with central India including Maharashtra and Madhya Pradesh, though early Buddhist texts describe Ashmaka as being further south, dakShiNApath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north. Recently in one of the scholarly studies based upon the astronomical readings in his works, it has been pointed out that Aryabhata's location may have been in Ponnani, Kerala .

However, it is fairly certain that at some point, he went to Kusumapura for higher studies, and that he lived here for some time. Bhāskara I (AD 629) identifies Kusumapura as Pataliputra (modern Patna). He lived there in the dying years of the Gupta empire, the time which is known as the golden age of India, when it was already under Hun attack in the Northeast, during the reign of Buddhagupta and some of the smaller kings before Vishnugupta.

Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature, and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary Varahamihira, as well as through later mathematicians and commentators including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta, and uses the midnight-day-reckoning, as opposed to sunrise in Aryabhatiya. This also contained a description of several astronomical instruments, the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semi-circle and circle shaped (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.

A third text that may have survived in Arabic translation is the Al ntf or Al-nanf, which claims to be a translation of Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the ninth c., it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

Aryabhatiya

Direct details of Aryabhata's work are therefore known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name; it is referred by his disciple Bhaskara I as Ashmakatantra or the treatise from the Ashmaka. It is also occasionally referred to as Arya-shatas-aShTa, lit., Aryabhata's 108, which is the number of verses in the text. It is written in the very terse style typical of the sutra literature, where each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The entire text consists of 108 verses, plus an introductory 13, the whole being divided into four pAdas or chapters:

1. Gitikapada: (13 verses) large units of time - kalpa, manvantra, yuga, which present a cosmology that differs from earlier texts such as Lagadha's Vedanga Jyotisha(ca. 1st c. BC). Also includes the table of sines (jya), given in a single verse. For the planetary revolutions during a mahayuga, the number of 4.32mn years is given.
2. Ganitapada (33 verses), covering mensuration (kShetra vyAvahAra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. Kalakriyapada (25 verses) : different units of time and method of determination of positions of planets for a given day. Calculations concerning the intercalary month (adhikamAsa), kShaya-tithis. Presents a seven-day week, with names for days of week.
4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon etc.

In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, ca. 600) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465).

Mathematics

Place Value system and zero

The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work. ; he certainly did not use the symbol, but the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients.

However, Aryabhata did not use the brahmi numerals; continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities (such as the table of sines) in a mnemonic form.

Pi as Irrational

Aryabhata worked on the approximation for Pi (π), and may have realized that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.
"Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached."

Aryabhata interpreted the word āsanna (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by Lambert).


After Aryabhatiya was translated into Arabic (ca. 820 AD) this approximation was mentioned in Al-Khwarizmi's book on algebra.

Mensuration and trigonometry

In Ganitapada 6, Aryabhata gives the area of triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: for a triangle, the result of a perpendicular with the half-side is the area. His great contribution to mensuration and trigonometry is used in the current international mathematics.

From "ardha-jya" to "sine"

Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". Because of simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba (after driven by the phonetic similarity). However, in Arabic writings, vowels are omitted and it got abbreviated to jb. When later writers realized that jb is an abbreviation of jiba, they substituted it back with jiab, means "cove" or "bay" (in Arabic, other than being merely a technical term, jiba is a meaningless word). Later in 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus (which has a same literal meaning of "cove" or "bay"). And after that, the sinus became sine in English, which is what the world now knows.

Indeterminate Equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. Here is an example from Bhaskara's commentary on Aryabhatiya: :

Find the number which gives 5 as the remainder when divided by 8; 4 as the remainder when divided by 9; and 1 as the remainder when divided by 7.

i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text Sulba Sutras, the more ancient parts of which may date back to 800 BCE. Aryabhata's method of solving such problems, called the kuṭṭaka (कूटटक) method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm, as elaborated by Bhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm.

The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

Astronomy

Aryabhata's system of astronomy was called the audAyaka system (days are reckoned from uday, dawn at lanka, equator). Some of his later writings on astronomy, which apparently proposed a second model (ardha-rAtrikA, midnight), are lost, but can be partly reconstructed from the discussion in Brahmagupta's khanDakhAdyaka. In some texts he seems to ascribe the apparent motions of the heavens to the earth's rotation.

Motions of the Solar System

Aryabhata appears to have believed that the earth rotates about its axis. This is made clear in the statement, referring to Lanka , which describes the movement of the stars as a relative motion caused by the rotation of the earth:

Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (i.e. on the equator) as moving exactly towards the West. [achalAni bhAni samapashchimagAni - golapAda.]

But the next verse describes the motion of the stars and planets as real movements: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.

Lanka (lit. Sri Lanka) is here a reference point on the equator, which was taken as the equivalent to the reference meridian for astronomical calculations.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle. The order of the planets in terms of distance from earth are taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.

The positions and periods of the planets was calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn move around the Earth at specific speeds representing each planet's motion through the zodiac. Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.

Eclipses

He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony where eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on earth. Thus the lunar eclipse occurs when the moon enters into the earth-shadow (verse gola.37), and discusses at length the size and extent of this earth-shadow (verses gola.38-48), and then the computation, and the size of the eclipsed part during eclipses. Subsequent Indian astronomers improved on these calculations, but his methods provided the core. This computational paradigm was so accurate that the 18th century scientist Guillaume le Gentil, during a visit to Pondicherry, found the Indian computations of the duration of the lunar eclipse of 1765-08-30 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds..

Aryabhata's computation of Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation was a significant improvement over the computation by the Greek mathematician, Eratosthenes (c. 200 BC), whose exact computation is not known in modern units but his estimate had an error of around 5-10%.

Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.

Heliocentrism

Āryabhata claimed that the Earth turns on its own axis and some elements of his planetary epicyclic models rotate at the same speed as the motion of the planet around the Sun. Thus it has been suggested that Āryabhata's calculations were based on an underlying heliocentric model in which the planets orbit the Sun. A detailed rebuttal to this heliocentric interpretation is in a review which describes B. L. van der Waerden's book as "show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata's description," although some concede that Āryabhata's system stems from an earlier heliocentric model of which he was unaware. It has even been claimed that he considered the planet's paths to be elliptical, although no primary evidence for this has been cited. Though Aristarchus of Samos (3rd century BC) and sometimes Heraclides of Pontus (4th century BC) are usually credited with knowing the heliocentric theory, the version of Greek astronomy known in ancient India, Paulisa Siddhanta (possibly by a Paul of Alexandria) makes no reference to a Heliocentric theory.

Legacy

Aryabhata's work was of great influence in the Indian astronomical tradition, and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (ca. 820), was particularly influential. Some of his results are cited by Al-Khwarizmi, and he is referred to by the 10th century Arabic scholar Al-Biruni, who states that Āryabhata's followers believed the Earth to rotate on its axis.

His definitions of sine, as well as cosine (kojya), versine (ukramajya), and inverse sine (otkram jya), influenced the birth of trigonometry. He was also the first to specify sine and versine (1 - cosx) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, the modern names "sine" and "cosine", are a mis-transcription of the words jya and kojya as introduced by Aryabhata. They were transcribed as jiba and kojiba in Arabic. They were then misinterpreted by Gerard of Cremona while translating an Arabic geometry text to Latin; he took jiba to be the Arabic word jaib, which means "fold in a garment", L. sinus (c.1150).

Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world, and were used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th c.), were translated into Latin as the Tables of Toledo (12th c.), and remained the most accurate Ephemeris used in Europe for centuries.

Calendric calculations worked out by Aryabhata and followers have been in continuous use in India for the practical purposes of fixing the Panchangam, or Hindu calendar, These were also transmitted to the Islamic world, and formed the basis for the Jalali calendar introduced 1073 by a group of astronomers including Omar Khayyam, versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The Jalali calendar determines its dates based on actual solar transit, as in Aryabhata (and earlier Siddhanta calendars). This type of calendar requires an Ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were lower in the Jalali calendar than in the Gregorian calendar.

India's first satellite Aryabhata, was named after him. The lunar crater Aryabhata is named in his honour. An Institute for conducting research in Astronomy, Astrophysics and atmospheric sciences has been named as Aryabhatta Research Institute of observational sciences (ARIES) near Nainital, India.

The interschool Aryabhatta Maths Competition is named after him.

ARYABATIA: in detail

Structure and style

The text is written in Sanskrit and structured into four section, overall covering 121 verses that describe different results using a mnemonic style typical of the Indian tradition.

33 verses are concerned with mathematical rules.

The four chapters are:

(i) the astronomical constants and the sine table (ii) mathematics required for computations (gaNitapāda) (iii) division of time and rules for computing the longitudes of planets using eccentrics and ellipses (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses (golādhyaya).

It is highly likely that the study of the Aryabhatiya was meant to be accompanied by the teachings of a well-versed tutor. While some of the verses have a logical flow, some don't and its lack of coherance makes it extremely difficult for a casual reader to follow.

Indian mathematical works often used word numerals before Aryabhata, but the Aryabhatiya is oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form words with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it. At the same time, this system of numeration allows for poetic license even in the author's choice of numbers. Cf. Āryabhaṭa numeration, the Sanskrit numerals.

Contents

Crowning glory of Aryabhatiya is the decimal place value notation without which mathematics, science and commerce would be impossible. Prior to Aryabhatta, Babylonians used 60 based place value notation which never gained momentum. Mathematics of Aryabhatta went to Europe through Arabs and was known as "Modus Indorum" or the method of the Indians. This method is none other than our arithmetic today.

The Aryabhatiya begins with an introduction called the "Dasagitika" or "Ten Giti Stanzas." This begins by paying tribute to Brahman, the "Cosmic spirit" in Hinduism. Next, Aryabhata lays out the numeration system used in the work. It includes a listing of astronomical constants and the sine table. The book then goes on to give an overview of Aryabhata's astronomical findings.

Most of the mathematics is contained in the next part, the "Ganitapada" or "Mathematics."

The next section is the "Kalakriya" or "The Reckoning of Time." In it, he divides up days, months, and years according to the movement of celestial bodies. He divides up history astrologically - it is from this exposition that historians deduced that the Aryabhatiya was written in 522 C.E. It also contains rules for computing the longitudes of planets using eccentrics and epicycles.

In the final section, the "Gola" or "The Sphere," Aryabhata goes into great detail describing the celestial relationship between the Earth and the cosmos. This section is noted for describing the rotation of the earth on its axis. It further uses the armillary sphere and details rules relating to problems of trigonometry and the computation of eclipses.

Significance The treatise uses a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle. It has also been interpreted as advocating Heliocentrism, where Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun (according to this view, it was heliocentric). Aryabhata asserted that the Moon and planets shine by reflected sunlight and that the orbits of the planets are ellipses. He also correctly explained the causes of eclipses of the Sun and the Moon. His value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the true value of 365 days 6 hours 9 minutes 10 seconds. In this book, the day was reckoned from one sunrise to the next, whereas in his "Āryabhata-siddhānta" he took the day from one midnight to another. There was also difference in some astronomical parameters. A close approximation to π is given as : "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, π ≈ 62832/20000 = 3.1416, correct to four rounded-off decimal places. Aryabhata was the first astronomer to make an attempt at measuring the Earth's circumference since Erastosthenes (circa 200 BC). Aryabhata accurately calculated the Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation remained the most accurate for over a thousand years. Aryabhata's methods of astronomical calculations have been in continuous use for practical purposes of fixing the Panchanga (Hindu calendar). Significant verses shulva-sUtras: form a shrauta part of kalpa vedAnga - nine texts - mathematically most imp - baudhAyana, Apastamba, and kAtyAyana shulvasUtra. dIrghasyAkShaNayA rajjuH pArshvamAnI tiryaDaM mAnI. cha yatpr^thagbhUte kurutastadubhayAM karoti. The diagonal of a rectangle produces both areas which its length and bread produce separately. samasya dvikaraNI. pramANaM tritIyena vardhayet tachchaturthAnAtma chatusastriMshenena savisheShaH. sqrt(2) = 1 + 1/3 + 1/(3.4) - 1(3.4.34) -- correct to 5 decimals = 1.41421569 chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAm AyutadvayaviShkambhasyAsanno vr^ttapariNahaH. [gaNita pAda, 10] Add 4 to 100, multiply by 8 and add to 62,000. This is approximately the circumference of a circle whose diamenter is 20,000. i.e. PI = 62,832 / 20,000 = 3.1416 correct to four places. Even more important however is the word "Asanna" - approximate, indicating an awareness that even this is an approximation. tribhujasya falasharIraM samadalakoTI bhujArdhasaMvargaH It depicts the area of a triangle. jyA = sine, koTijyA = cosine jyA tables : Circle circumference = minutes of arc = 360x60 = 21600. Gives radius R = radius of 3438; (exactly 21601.591) [ with pi = 3.1416, gives 21601.64] The R sine-differences (at intervals of 225 minutes of arc = 3:45deg), are given in an alphabetic code as 225,224,222,219.215,210,205, 199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7 which gives sines for 15 deg as sum of first four = 890 --> sin(15) = 890/3438 = 0.258871 vs. the correct value at 0.258819. sin(30) = 1719/3438 = 0.5 Expressed as the stanza, using the varga/avarga code: ka-M 1-5, ca-n~a: 6-10, Ta-Na 11-15, ta-na 16-20, pa-ma 21-25 the avargiya vyanjanas are: y = 30, r = 40, l=50, v=60, sh=70, Sh=80, s =90 and h=100 makhi (ma=25 + khi=2x100) bhakhi (24+200) fakhi (22+200) dhakhi (219) Nakhi 215, N~akhi 210, M~akhi 205, hasjha (h=100 + s=90+ jha=9) skaki (90+ ki=1x00 + ka=1) kiShga (1x100+80+3), shghaki, 70+4+100 kighva (100+4+60) ghlaki (4+50+100) kigra (100+3+40) hakya (100+1+30) dhaki (19+100) kicha (106) sga (93) shjha (79) Mva (5+60) kla (51) pta (21+16, could also have been chhya) fa (22) chha (7). makhi bhakhi dhakhi Nakhi N~akhi M~akhi hasjha 225, 224 222 219 215 210 205 skaki kiShga shghaki kighva ghlaki kigra hakya 199 191 183 174 164 154 143 dhaki kicha sga shjha Mva kla pta fa chha 119 106 93 79 65 51 37 22 7 given radius R = radius of 3438, these values give the Rxsin(theta) within one integer value; e.g. sine (15deg) = 225+224+222+219 = 890, modern value = 889.820. Both the choice of the radius based on the angle, and the 225 minutes of arc interpolation interval, are ideal for the table, better suited than the modern tables.

Translations

The Aryabhatiya was an extremely influential work as is exhibited by the fact that most notable Indian mathematicians after Aryabhata wrote commentaries on it. At least twelve notable commentaries were written for the Aryabhatiya ranging from the time he was still alive (c. 525) through 1900 ("Aryabhata I" 150-2). The commentators include Bhaskara and Brahmagupta among other notables.

The work was translated into Arabic around 820 by Al-Khwarizmi, whose On the Calculation with Hindu Numerals was in turn influential in the adoption of the Hindu-Arabic numerals in Europe from the 12th century.

Although the work was influential, there is no definitive English translation.

let talk about other great people who went BEYOND NATURE in the next post

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